In order theory, the strong set order is a partial order over the subsets of a lattice. It is widely used to study monotone comparative statics of parametrized optimization problems in economic theory and operations research.
The strong set order was first defined and used by Arthur F. Veinott in his unpublished lecture notes. It was later popularized by Donald M. Topkis, Paul Milgrom and Chris Shannon via their work on monotone comparative statics, in particular through Topkis' Theorem.
Definition
Given a lattice , consider its power set . The strong set order is a partial order on given by:
where and are respectively the join and meet of .
Examples and non-examples
Real numbers
Consider the real numbers with the usual order . Let and . Then
.
More generally, for any , we have
.
Euclidean space
Consider the Euclidean space with the usual pointwise order: . For , let
Then :. But now consider
Then , since , but
.
Power set
Consider the power set of the natural numbers under the set-inclusion order: . Let
,
.
It it not true that . Indeed, we have , , but
whence .
Applications
Topkis' Theorem
The strong set order is widely used in monotone comparative statics, in particular via Topkis' Theorem. Given a lattice , a poset , a constraint correspondence and a function , the theorem gives sufficient conditions for the correspondence
the be increasing in the strong set order, that is, for the statement
to hold.
Monotone selections
Increasigness in the strong set order can be used to obtain monotone selections from correspondences. Indeed, if is a poset and is a lattice, the correspondence is nonempty-valued and increasing in the strong set order, then a monotone selection from exists if any of the following hold:
has a minimal element for every (in particular, if is complete-lattice-valued). One can thus take : by putting . The same works if a maximal element is available.
is a sublattice of a finite product of chains. This covers, for example, .
is countable.
Zhou's Fixed-Point Theorem for correspondences
The strong set order can also be used for a generalization of Tarski's fixed-point theorem to correspondences known as Zhou's fixed-point theorem:
Theorem: let be a nonempty complete lattice and a nonempty-valued correspondence. If is increasing in the strong set order and is a subcomplete sublattice for all , then has a fixed point. Moreover, the set of such fixed points is a complete lattice.
Weak set order
The strong set order if often too restrictive for some applications, in particular because it assumes that the underlying space is a lattice. A useful weaker ordering which can be used on the subsets of any poset is the weak set order , defined by:
This order can moreover be broken down into two weaker orders: the upper and lower weak set orders , respectively defined by:
Clearly .
See also
Topkis's Theorem
Monotone comparative statics
References